We have the equation ax^2 + bx + c = 0 and it is given that the equation has equal roots.
If the equation has equal roots, it can be deduced that the difference of roots for the equation would be 0. For eg, if an equation has roots 3 and 3 (equal roots), the difference of roots will be 0. So, we'll apply this condition on the difference of roots formula
Difference of roots = √D / a = √(b²-4ac) / a
=> 0 = √(b²-4ac) / a
=> 0 = √(b²-4ac)
=> b² - 4ac = 0
=> 4ac = b^2
=> c = b^2/4a
Hence, if ax^2 + bx + c = 0 has equal roots, then c = b^2/4a
If the equation has equal roots, it can be deduced that the difference of roots for the equation would be 0. For eg, if an equation has roots 3 and 3 (equal roots), the difference of roots will be 0. So, we'll apply this condition on the difference of roots formula
Difference of roots = √D / a = √(b²-4ac) / a
=> 0 = √(b²-4ac) / a
=> 0 = √(b²-4ac)
=> b² - 4ac = 0
=> 4ac = b^2
=> c = b^2/4a
Hence, if ax^2 + bx + c = 0 has equal roots, then c = b^2/4a
Could you be much more clear about the answer....😕
ReplyDeleteSure. I am assuming you are unaware of the concept of difference of roots. If not, please be clear about what part of the solution you couldn't understand.
DeleteSo, if you know that for a standard quadratic equation,
ax^2 + bx + c = 0,
the sum of roots are -b/a
and the product of roots are c/a
We typically assume the roots to be alpha and beta. So we can write the same as
alpha + beta = -b/a ;
alpha * beta = c/a
We can use these two to derive (alpha - beta), which is the difference of the roots.
We have
alpha + beta = -b/a
squaring both the sides will give us
(alpha + beta)^2 = b^2/a^2
=> alpha^2 + beta^2 + 2.alpha.beta = b^2/a^2
Subtracting both sides of the equation by 4.alpha.beta will give us
=> alpha^2 + beta^2 + 2.alpha.beta - 4.alpha.beta = b^2/a^2 - 4.alpha.beta
=> alpha^2 + beta^2 - 2.alpha.beta = b^2/a^2 - 4.alpha.beta
=> (alpha - beta)^2 = b^2/a^2 - 4c/a (because alpha.beta = c/a)
=> (alpha - beta)^2 = (b^2 - 4ac)/a^2
=> alpha - beta = sqrt(b^2 -4ac) / a
We have used the same in a case where difference of the roots is 0, thereby the LHS of this very equation becomes 0.